In general, I argue that you shouldn't be able to re-use the grounds. That is, for any brewing session, your goal is to extract exactly what you want from the beans. If you do this optimally for your method and taste, there's nothing left in the beans that you want; re-extracting will give you a different result. The second brew might be drinkable and even tasty, but it will have a disproportionately small amount of stuff you might want, like caffeine and other flavour compounds.

That said, @Madmanta's answer makes a good point: it might work in some cases, such as Aeropress or espresso. But this suggests you might want to change other parameters of your brewing. If you can re-brew your coffee with a result that you like, you might be able to, for example, use less grounds. This saves money, which sounds like it's part of your goal. Alternatively, you could brew a larger cup of coffee with the same amount of beans, or increase the extraction time. In the case of AeroPress, for example, I find that many of the recipes are decidedly under-extracted, so this might be an explanation for why a second brew works for you; see this Q/A for what I mean by extraction in this case.

You could try this experimentally also: brew a cup of Aeropress or pour-over as you usually do, then brew another batch in the same way. For example, I find the last few drops of pour-over brew to be watery and off-tasting but the entire batch tastes better together than either of the two parts separately. If you like it, go for it. Or use half the amount of grounds the next time. :)

As a practical note, you might also consider simply to brew a larger batch of coffee all in one go. You'll get a slightly different product but, for example, I find that I use way less grounds for a 10-cup thermal carafe than for 10 individual cups.

In general, I argue that you shouldn't be able to re-use the grounds. That is, for any brewing session, your goal is to extract exactly what you want from the beans. If you do this optimally for your method and taste, there's nothing left in the beans that you want; re-extracting will give you a different result. The second brew might be drinkable and even tasty, but it will have a disproportionately small amount of stuff you might want, like caffeine and other flavour compounds.

That said, @Madmanta's answer makes a good point: it might work in some cases, such as Aeropress or espresso. But this suggests you might want to change other parameters of your brewing. If you can re-brew your coffee with a result that you like, you might be able to, for example, use less grounds. This saves money, which sounds like it's part of your goal. Alternatively, you could brew a larger cup of coffee with the same amount of beans, or increase the extraction time. In the case of AeroPress, for example, I find that many of the recipes are decidedly under-extracted, so this might be an explanation for why a second brew works for you; see this Q/A for what I mean by extraction in this case.

You could try this experimentally also: brew a cup of Aeropress or pour-over as you usually do, then brew another batch in the same way. For example, I find the last few drops of pour-over brew to be watery and off-tasting but the entire batch tastes better together than either of the two parts separately. If you like it, go for it. Or use half the amount of grounds the next time. :)

As a practical note, you might also consider simply to brew a larger batch of coffee all in one go. You'll get a slightly different product but, for example, I find that I use way less grounds for a 10-cup thermal carafe than for 10 individual cups.